The design of RF antennas can be exceedingly complex and mathematically and empirically intense due to the wide range of tradeoffs involving frequency response, sensitivity, directionality, polarizations, and so forth. Conventional antennas, such as open loops and parallel element arrays are limited in terms of applicability, such that, quite often, a particular geometry is relegated to a dedicated frequency band or direction.
It has been found that so-called fractal antennas offer certain advantages over conventional designs, including smaller size and desirable performance at multiple frequencies. In addition to greater frequency independence, such antennas afford enhanced radiation, since the often large number of sharp edges, corners, and discontinuities each act as points of electrical propagation or reception.
The term ‘fractal’ was coined by Benoit Mandelbrot in the mid-70s to describe a certain class of objects characterized in being self-similar and including multiple copies of the same shape but at different sizes or scales. Fractal patterns and multi-fractal patterns have by now been widely studied, and further information on fractal designs may be found in Frontiers in Electromagnetics, IEEE Press Series on Microwave Technology and RF, 2000, incorporated herein by reference.
Fractal antennae were first used to design multi-frequency arrays. The Sierpinski gasket antenna, which resembles a triangle packed with differently sized triangles of the same general orientation, was the first practical antenna to maintain performance at several (5) bands. Other fractal geometries used in antenna design include the Sierpinski carpet, which may be viewed as a square-within-a-square version of the Sierpinski gasket, as well as the snowflake or Koch curve, which has also been used in monopole form.
In designing an antenna based upon a folded or convoluted fractal-type geometry, the resonance frequencies may be a function of multiple parameters, including the shape of the structuring or replicated element, the size of the smallest element, and the number of scaling factors used simultaneously in the pattern. Despite the improved performance of antennas based upon fractal geometries, existing designs exhibit certain disadvantages. In particular, though self-similar, conventional fractals are based upon a heterogeneous reproduction of structuring elements limited to transformations in terms of rotation, translation, and scale, fractal structures utilize a subset of a Hutchinson operator W, wherein a regular shape, such as a triangle or square is iterated such that the same behavior may be obtained, albeit at multiple frequencies. Increased degrees of freedom are required to design structures with appropriate gain, beam patterns, polarization response, and other desirable characteristics.